Department of Mathematics
Math 01.561 School Mathematics from an Advanced Standpoint
School Mathematics from an Advanced Standpoint
This course is designed to develop a deeper understanding of mathematics
and a new
appreciation of its beauty, its logical structure and its applicability.
course will take into account not only the many interconnections among
school mathematics topics but also their relationship to higher mathematics.
At the end of this course, students will demonstrate the ability to:
• analyze alternative definitions, language, and algorithms for
mathematical ideas and concepts
• explain why concepts arose and how they changed over time
• perform a wide range of applications
• use calculator and computer approaches to problems
• analyze common high school math problems from a deeper mathematical
• expound on problems and proofs that can be extended and generalized
• provide connections between ideas in different branches of mathematics
• relate ideas studied in school mathematics to ideas studied
later in mathematics
B) Topical Outline
I. Algebra & Analysis with connections to Geometry
......• Real Numbers – decimal
representations of real numbers and periods of periodic decimals
......• Complex Numbers – the
geometry of complex number arithmetic
......• Functions – historical
evolution, problem analysis, properties, limiting behavior, fitting data
......• Equations – Isomorphism,
......• Integers & Polynomials
– Induction, recursion, FTA, bases, division, algorithm, Euclidean
.......• Number System Structures –
Modular arithmetic, integer congruence, number fields
II. Geometry with Connections to Algebra & Analysis
• Congruence – Euclid’s Elements, transformations,
• Distance & Similarity – Locus, distance on a sphere:
similarity of graphs, polygons, and arcs: geometric means
• Trigonometry – Modeling with trigonometric functions,
historical development, algebraic properties
• Area and Volume – Area formulas, area as probability,
• Axiomatics and Euclidean Geometry – Incidence Axioms,
Betweeness, Parallel Postulate, Euclid’s 5th, Cartesian coordinates
C) Course Texts
The following books may be used as a text for the course.
Beaumont & Pierce (1963). The Algebraic Foundations of Mathematics.
House & Coxfords (eds) (1995). Connecting Mathematics across the
.....1995 Yearbook of the NCTM.NCTM.
Ifah, Georges (2000). The Universal History of Numbers. Wiley.
Nelson, Roger (2000). Proofs Without Words II: More Exercises in Visual
Peressini, A. & Sherbert D. (1971). Topics in Modern Mathematics
.....Holt, Rinehardt & Winston.
Polya, George (1954). Induction and Analogy in Mathematics. Volume
I and II
Ribenboim, Paulo (2000). My Numbers, My Friends. Springer-Verlag.
Silvester, J (2001). Geometry: Ancient and Modern. Oxford University
Usiskin, Peressini, Marchisotto, & Stanley (2003). Mathematics for
.....Teachers. Person Education.